Solve 4x^4-25x^2y^2+6y^4

$$4 x ^ { 4 } - 25 x ^ { 2 } y ^ { 2 } + 6 y ^ { 4 }$$

Factor

$$\left(2x-y\right)\left(2x+y\right)\left(x^{2}-6y^{2}\right)$$

Solution Steps

Consider 4x^{4}-25x^{2}y^{2}+6y^{4} as a polynomial over variable x.

$$4x^{4}-25y^{2}x^{2}+6y^{4}$$

Find one factor of the form kx^{m}+n, where kx^{m} divides the monomial with the highest power 4x^{4} and n divides the constant factor 6y^{4}. One such factor is 4x^{2}-y^{2}. Factor the polynomial by dividing it by this factor.

$$\left(4x^{2}-y^{2}\right)\left(x^{2}-6y^{2}\right)$$

Consider 4x^{2}-y^{2}. Rewrite 4x^{2}-y^{2} as \left(2x\right)^{2}-y^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(2x-y\right)\left(2x+y\right)$$

Rewrite the complete factored expression.

$$\left(2x-y\right)\left(2x+y\right)\left(x^{2}-6y^{2}\right)$$

Evaluate

$$4x^{4}+6y^{4}-25\left(xy\right)^{2}$$